This work is devoted to the study of the initial boundary value problem for a general non isothermal model of capillary fluids derived by J. E Dunn and J. Serrin (1985) in [9, 16], which can be used as a phase transition model.We distinguish two cases, when the physical coefficients depend only on the density, and the general case. In the first case we can work in critical scaling spaces, and we prove global existence of solution and uniqueness for data close to a stable equilibrium. For general...

The existence of at least three weak solutions is established for a class of quasilinear elliptic equations involving the p(x)-biharmonic operator with Navier boundary value conditions. The proof is mainly based on a three critical points theorem due to B. Ricceri [Nonlinear Anal. 70 (2009), 3084-3089].

We study the following singular elliptic equation with critical exponent ⎧$-\Delta u=Q\left(x\right)u^{2*-1}+\lambda u^{-\gamma }$ in Ω, ⎨u > 0 in Ω, ⎩u = 0 on ∂Ω, where $\Omega \subset {ℝ}^N$ (N≥3) is a smooth bounded domain, and λ > 0, γ ∈ (0,1) are real parameters. Under appropriate assumptions on Q, by the constrained minimizer and perturbation methods, we obtain two positive solutions for all λ > 0 small enough.

We prove the existence of weak solutions for steady flows of electrorheological fluids with homogeneous Navier-slip type boundary conditions provided $p\left(x\right)>2n/(n+2)$. To prove this, we show Poincaré- and Korn-type inequalities, and then construct Lipschitz truncation functions preserving the zero normal component in variable exponent Sobolev spaces.

We prove existence of weak solutions to doubly degenerate diffusion equations $$\dot{u}={\Delta }_p{u}^{m-1}+f(m,p\ge 2)$$ by Faedo-Galerkin approximation for general domains and general nonlinearities. More precisely, we discuss the equation in an abstract setting, which allows to choose function spaces corresponding to bounded or unbounded domains $\Omega \subset {ℝ}^n$ with Dirichlet or Neumann boundary conditions. The function $f$ can be an inhomogeneity or a nonlinearity involving terms of the form $f\left(u\right)$ or $div\left(F\right(u\left)\right)$. In the appendix, an introduction to weak differentiability...

We prove the existence of a renormalized solution to a class of doubly nonlinear parabolic systems.

We investigate the existence of renormalized solutions for some nonlinear parabolic problems associated to equations of the form ⎧$\partial (e^{\beta u}-1)/\partial t-{div\left(\right|\nabla u|}^{p-2}\nabla u)+div(c(x,t){\left|u\right|}^{s-1}u)+b(x,t){\left|\nabla u\right|}^r=f$ in Q = Ω×(0,T), ⎨ u(x,t) = 0 on ∂Ω ×(0,T), ⎩ $(e^{\beta u}-1)(x,0)=(e^{\beta u₀}-1)\left(x\right)$ in Ω. with s = (N+2)/(N+p) (p-1), $c(x,t)\in {\left(L^{\tau }\left(QT\right)\right)}^N$, τ = (N+p)/(p-1), r = (N(p-1) + p)/(N+2), $b(x,t)\in L^{N+2,1}\left(QT\right)$ and f ∈ L¹(Q).

Let p,q,n be natural numbers such that p+q = n. Let be either ℂ, the complex numbers field, or ℍ, the quaternionic division algebra. We consider the Heisenberg group N(p,q,) defined ⁿ × ℑ , with group law given by (v,ζ)(v’,ζ’) = (v + v’, ζ + ζ’- 1/2 ℑ B(v,v’)), where $B(v,w)={\sum }_{j=1}^p{v}_j\overline{w_j}-{\sum }_{j=p+1}^n{v}_j\overline{w_j}$. Let U(p,q,) be the group of n × n matrices with coefficients in that leave the form B invariant. We compute explicit fundamental solutions of some second order differential operators on N(p,q,) which are canonically associated to...

Let $X$ be a complex manifold, $S$ a generic submanifold of $X^{\mathbb{R}}$, the real underlying manifold to $X$. Let $\mathrm{\Omega }$ be an open subset of $S$ with $\partial \mathrm{\Omega }$ analytic, $Y$ a complexification of $S$. We first recall the notion of $\mathrm{\Omega }$-tuboid of $X$ and of $Y$ and then give a relation between; we then give the corresponding result in terms of microfunctions at the boundary. We relate the regularity at the boundary for ${\overline{\partial }}_b$ to the extendability of $CR$ functions on $\mathrm{\Omega }$ to $\mathrm{\Omega }$-tuboids of $X$. Next, if $X$ has complex dimension 2, we give results on extension...